3.575 \(\int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=436 \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}+\frac {\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]

[Out]

1/512*(-a*d+b*c)^4*(5*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(1
1/2)/c^(7/2)-1/60*(5*a*d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/x^5-1/6*(d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6+1/160*(-5*
a^2*d^2-6*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/x^4-1/960*(5*a^3*d^3+51*a^2*b*c*d^2-61*a*b^2*c^2*
d+21*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c/x^3+1/3840*(25*a^4*d^4-20*a^3*b*c*d^3+262*a^2*b^2*c^2*d^2-308*
a*b^3*c^3*d+105*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x^2-1/7680*(75*a^5*d^5-65*a^4*b*c*d^4-90*a^3*b^2*
c^2*d^3+838*a^2*b^3*c^3*d^2-945*a*b^4*c^4*d+315*b^5*c^5)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3/x

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Rubi [A]  time = 0.47, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {97, 149, 151, 12, 93, 208} \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (51 a^2 b c d^2+5 a^3 d^3-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}+\frac {\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]

[Out]

((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(160*a^2*x^4) - ((21*b^3*c^3 - 61*a*b^2*c^2*
d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c*x^3) + ((105*b^4*c^4 - 308*a*b^3*c^3*d
+ 262*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3840*a^4*c^2*x^2) - ((315*
b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5*d^5)*Sqrt[a + b
*x]*Sqrt[c + d*x])/(7680*a^5*c^3*x) - ((b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(60*a*x^5) - (Sqrt[a + b*x
]*(c + d*x)^(5/2))/(6*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(11/2)*c^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {1}{6} \int \frac {(c+d x)^{3/2} \left (\frac {1}{2} (b c+5 a d)+3 b d x\right )}{x^6 \sqrt {a+b x}} \, dx\\ &=-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {\int \frac {\sqrt {c+d x} \left (-\frac {3}{4} \left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right )-\frac {3}{2} b d (b c-5 a d) x\right )}{x^5 \sqrt {a+b x}} \, dx}{30 a}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {\int \frac {\frac {3}{8} \left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right )+\frac {3}{4} b d \left (9 b^2 c^2-26 a b c d+25 a^2 d^2\right ) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^2}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\int \frac {\frac {3}{16} \left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right )+\frac {3}{4} b d \left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{360 a^3 c}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {\int \frac {\frac {3}{32} \left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right )+\frac {3}{16} b d \left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{720 a^4 c^2}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt {a+b x} \sqrt {c+d x}}{7680 a^5 c^3 x}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\int \frac {45 (b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )}{64 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{720 a^5 c^3}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt {a+b x} \sqrt {c+d x}}{7680 a^5 c^3 x}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 a^5 c^3}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt {a+b x} \sqrt {c+d x}}{7680 a^5 c^3 x}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 a^5 c^3}\\ &=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt {a+b x} \sqrt {c+d x}}{7680 a^5 c^3 x}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{11/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 267, normalized size = 0.61 \[ \frac {\frac {\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \left (\frac {x (b c-a d) \left (\frac {5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}-8 \sqrt {a+b x} (c+d x)^{5/2}\right )}{a}-48 \sqrt {a+b x} (c+d x)^{7/2}\right )}{64 c x^4}-\frac {20 a c (a+b x)^{3/2} (c+d x)^{7/2}}{x^6}+\frac {2 (a+b x)^{3/2} (c+d x)^{7/2} (5 a d+9 b c)}{x^5}}{120 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]

[Out]

((-20*a*c*(a + b*x)^(3/2)*(c + d*x)^(7/2))/x^6 + (2*(9*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/x^5 + ((2
1*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*(-48*Sqrt[a + b*x]*(c + d*x)^(7/2) + ((b*c - a*d)*x*(-8*Sqrt[a + b*x]*(c +
d*x)^(5/2) + (5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b
*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*Sqrt[c])))/a))/(64*c*x^4))
/(120*a^2*c^2)

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fricas [A]  time = 25.26, size = 918, normalized size = 2.11 \[ \left [\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, a^{6} d^{6}\right )} \sqrt {a c} x^{6} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (1280 \, a^{6} c^{6} + {\left (315 \, a b^{5} c^{6} - 945 \, a^{2} b^{4} c^{5} d + 838 \, a^{3} b^{3} c^{4} d^{2} - 90 \, a^{4} b^{2} c^{3} d^{3} - 65 \, a^{5} b c^{2} d^{4} + 75 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (105 \, a^{2} b^{4} c^{6} - 308 \, a^{3} b^{3} c^{5} d + 262 \, a^{4} b^{2} c^{4} d^{2} - 20 \, a^{5} b c^{3} d^{3} + 25 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (21 \, a^{3} b^{3} c^{6} - 61 \, a^{4} b^{2} c^{5} d + 51 \, a^{5} b c^{4} d^{2} + 5 \, a^{6} c^{3} d^{3}\right )} x^{3} - 16 \, {\left (9 \, a^{4} b^{2} c^{6} - 26 \, a^{5} b c^{5} d - 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, a^{6} c^{4} x^{6}}, -\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, a^{6} d^{6}\right )} \sqrt {-a c} x^{6} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (1280 \, a^{6} c^{6} + {\left (315 \, a b^{5} c^{6} - 945 \, a^{2} b^{4} c^{5} d + 838 \, a^{3} b^{3} c^{4} d^{2} - 90 \, a^{4} b^{2} c^{3} d^{3} - 65 \, a^{5} b c^{2} d^{4} + 75 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (105 \, a^{2} b^{4} c^{6} - 308 \, a^{3} b^{3} c^{5} d + 262 \, a^{4} b^{2} c^{4} d^{2} - 20 \, a^{5} b c^{3} d^{3} + 25 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (21 \, a^{3} b^{3} c^{6} - 61 \, a^{4} b^{2} c^{5} d + 51 \, a^{5} b c^{4} d^{2} + 5 \, a^{6} c^{3} d^{3}\right )} x^{3} - 16 \, {\left (9 \, a^{4} b^{2} c^{6} - 26 \, a^{5} b c^{5} d - 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, a^{6} c^{4} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

[1/30720*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^
5*b*c*d^5 + 5*a^6*d^6)*sqrt(a*c)*x^6*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c +
a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(1280*a^6*c^6 + (315*a*b^5*c
^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^4*d^2 - 90*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(
105*a^2*b^4*c^6 - 308*a^3*b^3*c^5*d + 262*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 25*a^6*c^2*d^4)*x^4 + 8*(21*a^3
*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*c^4*d^2 + 5*a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135
*a^6*c^4*d^2)*x^2 + 128*(a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^4*x^6), -1/15360*(15
*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 +
5*a^6*d^6)*sqrt(-a*c)*x^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x
^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(1280*a^6*c^6 + (315*a*b^5*c^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^
4*d^2 - 90*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(105*a^2*b^4*c^6 - 308*a^3*b^3*c^5*d + 2
62*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 25*a^6*c^2*d^4)*x^4 + 8*(21*a^3*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*
c^4*d^2 + 5*a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135*a^6*c^4*d^2)*x^2 + 128*(a^5*b*c^6 + 25
*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^4*x^6)]

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giac [B]  time = 45.07, size = 8502, normalized size = 19.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/7680*(15*(21*sqrt(b*d)*b^7*c^6*abs(b) - 70*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 75*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b
) - 20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b) - 5*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) - 6*sqrt(b*d)*a^5*b^2*c*d^5*abs(b
) + 5*sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^5*b*c^3) - 2*(315*sqrt(b*d)*b^29*c^17*abs(b) - 4725*
sqrt(b*d)*a*b^28*c^16*d*abs(b) + 32968*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 141816*sqrt(b*d)*a^3*b^26*c^14*d^3
*abs(b) + 420148*sqrt(b*d)*a^4*b^25*c^13*d^4*abs(b) - 906700*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 1468920*sqrt
(b*d)*a^6*b^23*c^11*d^6*abs(b) - 1811656*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 1701282*sqrt(b*d)*a^8*b^21*c^9*d
^8*abs(b) - 1195326*sqrt(b*d)*a^9*b^20*c^8*d^9*abs(b) + 595320*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) - 174280*sq
rt(b*d)*a^11*b^18*c^6*d^11*abs(b) - 4812*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) + 34484*sqrt(b*d)*a^13*b^16*c^4*d
^13*abs(b) - 18872*sqrt(b*d)*a^14*b^15*c^3*d^14*abs(b) + 5640*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) - 965*sqrt(b
*d)*a^16*b^13*c*d^16*abs(b) + 75*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 3465*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^27*c^16*abs(b) + 41160*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a*b^26*c^15*d*abs(b) - 218616*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^2*a^2*b^25*c^14*d^2*abs(b) + 675384*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^2*a^3*b^24*c^13*d^3*abs(b) - 1305900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^4*b^23*c^12*d^4*abs(b) + 1515720*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^5*b^22*c^11*d^5*abs(b) - 667560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^6*b^21*c^10*d^6*abs(b) - 987528*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^7*b^20*c^9*d^7*abs(b) + 2269242*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^8*b^19*c^8*d^8*abs(b) - 2304360*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^9*b^18*c^7*d^9*abs(b) + 1407480*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^10*b^17*c^6*d^10*abs(b) - 485400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^11*b^16*c^5*d^11*abs(b) + 27444*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^12*b^15*c^4*d^12*abs(b) + 62424*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^13*b^14*c^3*d^13*abs(b) - 33240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^14*b^13*c^2*d^14*abs(b) + 8040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^15*b^12*c*d^15*abs(b) - 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^2*a^16*b^11*d^16*abs(b) + 17325*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^4*b^25*c^15*abs(b) - 158865*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a*b^24*c^14*d*abs(b) + 620433*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^4*a^2*b^23*c^13*d^2*abs(b) - 1310205*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^4*a^3*b^22*c^12*d^3*abs(b) + 1538025*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^4*b^21*c^11*d^4*abs(b) - 890925*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^5*b^20*c^10*d^5*abs(b) + 335565*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^6*b^19*c^9*d^6*abs(b) - 1032441*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^7*b^18*c^8*d^7*abs(b) + 2299575*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^8*b^17*c^7*d^8*abs(b) - 2523075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^9*b^16*c^6*d^9*abs(b) + 1534275*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^10*b^15*c^5*d^10*abs(b) - 457575*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^11*b^14*c^4*d^11*abs(b) - 24717*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^12*b^13*c^3*d^12*abs(b) + 77505*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^13*b^12*c^2*d^13*abs(b) - 29025*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^14*b^11*c*d^14*abs(b) + 4125*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^4*a^15*b^10*d^15*abs(b) - 51975*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^6*b^23*c^14*abs(b) + 357210*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^2
2*c^13*d*abs(b) - 984045*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^21*
c^12*d^2*abs(b) + 1332900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^20
*c^11*d^3*abs(b) - 847775*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^19
*c^10*d^4*abs(b) + 209670*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^18
*c^9*d^5*abs(b) - 104205*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^17*
c^8*d^6*abs(b) - 150920*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^16*c
^7*d^7*abs(b) + 879075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^15*c^
6*d^8*abs(b) - 1160250*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^9*b^14*c^
5*d^9*abs(b) + 673105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^10*b^13*c^
4*d^10*abs(b) - 114780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^11*b^12*c
^3*d^11*abs(b) - 84285*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^12*b^11*c
^2*d^12*abs(b) + 58650*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^13*b^10*c
*d^13*abs(b) - 12375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^14*b^9*d^14
*abs(b) + 103950*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^21*c^13*abs(b)
- 518490*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^20*c^12*d*abs(b) + 95
7780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^19*c^11*d^2*abs(b) - 74
8620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^3*b^18*c^10*d^3*abs(b) + 16
5930*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^4*b^17*c^9*d^4*abs(b) + 224
610*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^5*b^16*c^8*d^5*abs(b) - 7544
40*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^15*c^7*d^6*abs(b) + 10906
80*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^14*c^6*d^7*abs(b) - 55347
0*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^8*b^13*c^5*d^8*abs(b) - 100470
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^9*b^12*c^4*d^9*abs(b) + 150900*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^10*b^11*c^3*d^10*abs(b) + 27540*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^11*b^10*c^2*d^11*abs(b) - 70650*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^12*b^9*c*d^12*abs(b) + 24750*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^13*b^8*d^13*abs(b) - 145530*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^19*c^12*abs(b) + 511560*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^18*c^11*d*abs(b) - 589428*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^17*c^10*d^2*abs(b) + 188136*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^16*c^9*d^3*abs(b) + 38202*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*b^15*c^8*d^4*abs(b) + 147600*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^5*b^14*c^7*d^5*abs(b) - 651480*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^13*c^6*d^6*abs(b) + 797712*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^7*b^12*c^5*d^7*abs(b) - 184614*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^8*b^11*c^4*d^8*abs(b) - 156888*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^9*b^10*c^3*d^9*abs(b) + 31500*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^10*b^9*c^2*d^10*abs(b) + 47880*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^11*b^8*c*d^11*abs(b) - 34650*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^12*b^7*d^12*abs(b) + 145530*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^17*c^11*abs(b) - 357210*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^16*c^10*d*abs(b) + 215166*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^15*c^9*d^2*abs(b) + 34146*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^14*c^8*d^3*abs(b) + 420*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^4*b^13*c^7*d^4*abs(b) + 47900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^5*b^12*c^6*d^5*abs(b) + 70620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^6*b^11*c^5*d^6*abs(b) - 241564*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^7*b^10*c^4*d^7*abs(b) + 365826*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^8*b^9*c^3*d^8*abs(b) - 42210*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^9*b^8*c^2*d^9*abs(b) - 11130*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^12*a^10*b^7*c*d^10*abs(b) + 34650*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^12*a^11*b^6*d^11*abs(b) - 103950*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^14*b^15*c^10*abs(b) + 187740*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^14*a*b^14*c^9*d*abs(b) - 18198*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^14*a^2*b^13*c^8*d^2*abs(b) - 39600*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^14*a^3*b^12*c^7*d^3*abs(b) - 27900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^14*a^4*b^11*c^6*d^4*abs(b) - 231000*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^14*a^5*b^10*c^5*d^5*abs(b) + 334020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^14*a^6*b^9*c^4*d^6*abs(b) - 489648*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^14*a^7*b^8*c^3*d^7*abs(b) + 28170*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^14*a^8*b^7*c^2*d^8*abs(b) - 8100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^14*a^9*b^6*c*d^9*abs(b) - 24750*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^14*a^10*b^5*d^10*abs(b) + 51975*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16
*b^13*c^9*abs(b) - 81585*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a*b^12*c
^8*d*abs(b) - 28260*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^2*b^11*c^7*
d^2*abs(b) + 24300*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^3*b^10*c^6*d
^3*abs(b) + 54450*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^4*b^9*c^5*d^4
*abs(b) - 61230*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^5*b^8*c^4*d^5*a
bs(b) + 283500*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^6*b^7*c^3*d^6*ab
s(b) - 16740*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^7*b^6*c^2*d^7*abs(
b) + 6975*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^8*b^5*c*d^8*abs(b) +
12375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^9*b^4*d^9*abs(b) - 17325*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*b^11*c^8*abs(b) + 30240*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a*b^10*c^7*d*abs(b) + 12500*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^2*b^9*c^6*d^2*abs(b) - 24000*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^3*b^8*c^5*d^3*abs(b) - 31550*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^4*b^7*c^4*d^4*abs(b) - 53920*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^5*b^6*c^3*d^5*abs(b) + 7860*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^6*b^5*c^2*d^6*abs(b) - 1600*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^7*b^4*c*d^7*abs(b) - 4125*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^8*b^3*d^8*abs(b) + 3465*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*b^9*c^7*abs(b) - 8085*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^20*a*b^8*c^6*d*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^20*a^2*b^7*c^5*d^2*abs(b) + 9075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^20*a^3*b^6*c^4*d^3*abs(b) - 4125*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^20*a^4*b^5*c^3*d^4*abs(b) - 1815*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^20*a^5*b^4*c^2*d^5*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^20*a^6*b^3*c*d^6*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20
*a^7*b^2*d^7*abs(b) - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*b^7*c^6
*abs(b) + 1050*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a*b^6*c^5*d*abs(b)
- 1125*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^2*b^5*c^4*d^2*abs(b) +
300*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^3*b^4*c^3*d^3*abs(b) + 75*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^4*b^3*c^2*d^4*abs(b) + 90*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^5*b^2*c*d^5*abs(b) - 75*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^6*b*d^6*abs(b))/((b^4*c^2 - 2*a*b^3*c*d + a^
2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4)^6*a^5*c^3))/b

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maple [B]  time = 0.04, size = 1271, normalized size = 2.92 \[ \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (75 a^{6} d^{6} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-90 a^{5} b c \,d^{5} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-75 a^{4} b^{2} c^{2} d^{4} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-300 a^{3} b^{3} c^{3} d^{3} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+1125 a^{2} b^{4} c^{4} d^{2} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-1050 a \,b^{5} c^{5} d \,x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+315 b^{6} c^{6} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-150 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5} x^{5}+130 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4} x^{5}+180 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3} x^{5}-1676 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2} x^{5}+1890 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d \,x^{5}-630 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{5} x^{5}+100 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c \,d^{4} x^{4}-80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{2} d^{3} x^{4}+1048 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{3} d^{2} x^{4}-1232 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{4} d \,x^{4}+420 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{5} x^{4}-80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{2} d^{3} x^{3}-816 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{3} d^{2} x^{3}+976 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{4} d \,x^{3}-336 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{5} x^{3}-4320 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{3} d^{2} x^{2}-832 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{4} d \,x^{2}+288 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{5} x^{2}-6400 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{4} d x -256 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{5} x -2560 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{5} c^{5}\right )}{15360 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{5} c^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x)

[Out]

1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3*(1125*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2))/x)*x^6*a^2*b^4*c^4*d^2-150*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^5*d^5+75*ln((a*d*x+b*c*x
+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^6*a^6*d^6+315*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^6*b^6*c^6-2560*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c^5*(a*c)^(1/2)-630*(a
*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*b^5*c^5-1050*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2))/x)*x^6*a*b^5*c^5*d+100*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^5*c*d^4+420*(a*c)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^4*c^5-80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^5*c^2*
d^3-336*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^3*c^5-4320*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*x^2*a^5*c^3*d^2+288*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b^2*c^5-6400*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*c^4*d-256*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*c^5-90*ln((a*d*x
+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^6*a^5*b*c*d^5-75*ln((a*d*x+b*c*x+2*a*c+2*(a*c
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^6*a^4*b^2*c^2*d^4-300*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^6*a^3*b^3*c^3*d^3+130*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^4*b*c*
d^4+180*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^3*b^2*c^2*d^3-1676*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*x^5*a^2*b^3*c^3*d^2+1890*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a*b^4*c^4*d-832*(a*c)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^4*b*c^4*d-80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*b*c^2
*d^3+1048*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b^2*c^3*d^2-1232*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*x^4*a^2*b^3*c^4*d-816*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^4*b*c^3*d^2+976*(a*c)^(1/
2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b^2*c^4*d)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^6/(a*c)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^7,x)

[Out]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^7, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**7,x)

[Out]

Timed out

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